Some Basic Theories Of L-fuzzy Normed Linear Space

L-fuzzy normed linear space is a natural generalization of fuzzy normed linear space and classical normed linear space. This paper bends itself to study the basic theory of L-fuzzy normed linear spaces. The main contents involve the L-topological structure of L-fuzzy normed linear space and its basic properties, the completion of L-fuzzy normed linear space, the boundedness of L-fuzzy sets and L-fuzzy linear order-homomorphisms and the space ofφ-variable basis powerset bounded linear operators. The paper is organized as follows:In Chapter 1, as preliminaries, we introduce the basic concepts and results about fuzzy lattice, L-fuzzy set, L-fuzzy linear order-homomorphism, L-topological space and L-topological vector space etc.In Chapter 2, we investigate the L-topological structure of L-fuzzy normed linear space, and prove that the L-fuzzy normed linear space is a special type of Hausdorff locally convex and locally bounded L-topological vector space by a new way. We give a characterization for the convergence of a sequence of L-fuzzy points in L-fuzzy normed linear space and its application. Moreover, we discuss the equivalence of L-fuzzy norms.In Chapter 3, we prove the completion theorem of L-fuzzy normed linear space, i.e. for every L-fuzzy normed linear space (L~X, || ? ||), always there exists a L-fuzzy Banach space (L~X, |(?)| ? |||) such that (L~X, ||·||) is isometrically isomorphic to a subspace (L~w, ||| ? |||) of L~X which is uniformly dense for all stratum in (L~X, ||| ? |||), and the completion space of (L~X, ||·||) is unique in the sense of isometrical isomorphism. In Chapter 4, we study the characterizations for the bounded L-fuzzy set and bounded L-fuzzy linear order-homomorphism. We give a definition of L-fuzzy norm of a L-fuzzy linear order-homomorphism. On the basis of this, we discuss the boundedness and the continuity of L-fuzzy linear order-homomorphisms.In Chapter 5, by using the L-fuzzy norm of L-fuzzy linear order-homomorphism, the space ofφ-variable basis powerset bounded linear operators is investigated. We prove that this space is an L-fuzzy normed space for the L-fuzzy norm of L-fuzzy linear order-homomorphism. Moreover, we also give a necessary and sufficient condition for the space ofφ-variable basis powerset bounded linear operators to be complete.